Seminars and Colloquia by Series

Ramanujan and Expander Graphs

Series
Stelson Lecture Series
Time
Thursday, October 24, 2024 - 16:30 for 1 hour (actually 50 minutes)
Location
Bill Moore Student Success Center, Press Rooms A & B
Speaker
Peter Sarnak – Princeton University –

Expander graphs are highly connected sparse graphs. They have wide theoretical and practical applications in Computer Science and Engineering. Ramanujan Graphs are optimal expanders and as the name suggests they are constructed number theoretically. We review their construction as well more recent constructions that use statistical physics. We highlight some recent applications in the reverse direction where combinatorial ideas are combined with arithmetical ones to establish expansion of graphs arising in diophantine analysis.

From Coffee to Mathematics: Making Connections and Finding Unexpected Links

Series
Stelson Lecture Series
Time
Thursday, March 7, 2024 - 16:30 for 1 hour (actually 50 minutes)
Location
Howey-Physics L3
Speaker
Hugo Duminil-Copin – Université de Genève and IHES Université Paris-Saclay

The game of HEX has deep mathematical underpinnings despite its simple rules.  What could this game possibly have to do with coffee?!  And how does that connection, once identified, lead to consideration of ferromagnetism and even to the melting polar ice caps?  Join Hugo Duminil-Copin, Professor of Mathematics at IHES and the University of Geneva, for an exploration of the way in which mathematical thinking can help us make some truly surprising connections.

Lyapunov exponents, Schrödinger cocycles, and Avila’s global theory

Series
Stelson Lecture Series
Time
Tuesday, March 14, 2023 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Wilhelm Schlag – Yale University –

Please Note: Mathematics lecture

 In the 1950s Phil Anderson made a prediction about the effect of random impurities on the conductivity properties of a crystal. Mathematically, these questions amount to the study of solutions of differential or difference equations and the associated spectral theory of self-adjoint operators obtained from an ergodic process. With the arrival of quasicrystals, in addition to random models, nonrandom lattice models such as those generated by irrational rotations or skew-rotations on tori have been studied over the past 30 years. 

By now, an extensive mathematical theory has developed around Anderson’s predictions, with several questions remaining open. This talk will attempt to survey certain aspects of the field, with an emphasis on the theory of SL(2,R) cocycles with an irrational or  Diophantine  rotation on the circle as base dynamics. In this setting, Artur Avila discovered about a decade ago that the Lyapunov exponent is piecewise affine in the imaginary direction after complexification of the circle. In fact, the slopes of these affine functions are integer valued. This is easy to see in the uniformly hyperbolic case, which is equivalent to energies falling into the gaps of the spectrum, due to the winding number of the complexified Lyapunov exponent. Remarkably, this property persists also in the non-uniformly hyperbolic case, i.e., on the spectrum of the Schrödinger operator. This requires a delicate continuity property of the Lyapunov exponent in both energy and frequency. Avila built his global theory (Acta Math. 2015) on this quantization property. I will present some recent results with Rui HAN (Louisiana) connecting Avila’s notion of  acceleration (the slope of the complexified Lyapunov exponent in the imaginary direction) to the number of zeros of the determinants of  finite volume Hamiltonians relative to the complex toral variable. This connection allows one to answer questions arising in the supercritical case of Avila’s global theory concerning the measure of the second stratum, Anderson localization on this stratum, as well as settle a conjecture on the Hölder regularity of the integrated density of states.

Nonlinear waves, spectra, and dynamics in infinite dimensions

Series
Stelson Lecture Series
Time
Friday, March 10, 2023 - 16:00 for 1 hour (actually 50 minutes)
Location
Klaus lecture auditorium 1443
Speaker
Wilhelm Schlag – Yale University –

Please Note: General audience lecture

Waves are ubiquitous in nature. Some wave phenomena are conspicuous, most notably in elastic objects, and in bodies of water. In electro-dynamics, quantum mechanics, and gravity, waves play a fundamental role but are much more difficult to find. Over the past centuries, major scientific breakthroughs have been associated with the discovery of hidden wave phenomena in nature. Engineering has enabled our modern information based society by developing sophisticated methods which allow us to harness wave propagation. Seismic exploration relies on wave scattering in the discovery of natural resources. Medicine depends heavily on wave-based imaging technology such as MRI and CAT scans.

 

Mathematics has played a major role in the understanding of wave propagation, and its many intricate phenomena including reflection, diffraction, and refraction. In its most basic form, the wave equation is a linear partial differential equation (PDE). However, modern science and engineering rely heavily on nonlinear PDEs which can exhibit many surprising and delicate properties. Mathematical analysis continues to evolve rapidly driven in part by the many open questions surrounding nonlinear PDEs and their solutions. This talk will survey some of the mathematics involved in our understanding of waves, both linear and nonlinear.

Cryptography: From ancient times to a post-quantum age

Series
Stelson Lecture Series
Time
Thursday, March 1, 2018 - 18:00 for 1 hour (actually 50 minutes)
Location
Klaus Lecture Auditorium 1443
Speaker
Jill Pipher – Brown University
How is it possible to send encrypted information across an insecure channel (like the internet) so that only the intended recipient can decode it, without sharing the secret key in advance? In 1976, well before this question arose, a new mathematical theory of encryption (public-key cryptography) was invented by Diffie and Hellman, which made digital commerce and finance possible. The technology advances of the last twenty years bring new and urgent problems, including the need to compute on encrypted data in the cloud and to have cryptography that can withstand the speed-ups of quantum computers. In this lecture, we will discuss some of the history of cryptography, as well as some of the latest ideas in "lattice" cryptography which appear to be quantum resistant and efficient.

How quantum theory and statistical mechanics gave a polynomial of knots

Series
Stelson Lecture Series
Time
Thursday, September 25, 2014 - 16:35 for 1 hour (actually 50 minutes)
Location
Clary Theater, Student Success Center
Speaker
Vaughan Jones – University of Vanderbilt –
We will see how a result in von Neumann algebras (a theory developed by von Neumann to give themathematical framework for quantum physics) gave rise, rather serendipitously, to an elementary but very usefulinvariant in the theory of ordinary knots in threel space. Then we'll look at some subsequent developments of the theory, and talk about a thorny problem which remains open.

Riemann, Boltzmann and Kantorovich go to a party

Series
Stelson Lecture Series
Time
Monday, April 22, 2013 - 16:00 for 1.5 hours (actually 80 minutes)
Location
Klaus 1116
Speaker
Cedric Villani – Institut Henri Poincare, CNRS/UPMC

Please Note: General Audience Lecture. Reception to follow in Klaus Atrium.

This talk is the story of an encounter of three distinct fields: non-Euclidean geometry, gas dynamics and economics. Some of the most fundamental mathematical tools behind these theories appear to have a close connection, which was revealed around the turn of the 21st century, and has developed strikingly since then.

From Optimal Transport to Fully Nonlinear PDE to Regularity to Nonsmooth Geometry

Series
Stelson Lecture Series
Time
Friday, April 19, 2013 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Cedric Villani – Institut Henri Poincare (CNRS/UPMC)

Please Note: Mathematics Audience Lecture

This talk explains how the solution to a regularity/geometry problem arising from a question of optimization has led to unexpected new results in the well-established field of the analysis of cut loci.

PhaseLift: Exact Phase Retrieval via Convex Programming

Series
Stelson Lecture Series
Time
Tuesday, September 11, 2012 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Emmanuel Candes – Departments of Mathematics and Statistics, Stanford University

Please Note: Mathematics lecture

This talks introduces a novel framework for phase retrieval, a problem which arises in X-ray crystallography, diffraction imaging, astronomical imaging and many other applications. Our approach combines multiple structured illuminations together with ideas from convex programming to recover the phase from intensity measurements, typically from the modulus of the diffracted wave. We demonstrate empirically that any complex-valued object can be recovered from the knowledge of the magnitude of just a few diffracted patterns by solving a simple convex optimization problem inspired by the recent literature on matrix completion. More importantly, we also demonstrate that our noise-aware algorithms are stable in the sense that the reconstruction degrades gracefully as the signal-to-noise ratio decreases. Finally, we present some novel theory showing that our entire approach may be provably surprisingly effective.

Robust principal component analysis? Some theory and some applications

Series
Stelson Lecture Series
Time
Monday, September 10, 2012 - 16:25 for 1 hour (actually 50 minutes)
Location
Clough Commons Room 144
Speaker
Emmanuel Candes – Stanford University

Please Note: General audience lecture

This talk is about a curious phenomenon. Suppose we have a data matrix, which is the superposition of a low-rank component and a sparse component. Can we recover each component individually? We prove that under some suitable assumptions, it is possible to recover both the low-rank and the sparse components exactly by solving a very convenient convex program. This suggests the possibility of a principled approach to robust principal component analysis since our methodology and results assert that one can recover the principal components of a data matrix even though a positive fraction of its entries are arbitrarily corrupted. This extends to the situation where a fraction of the entries are missing as well. In the second part of the talk, we present applications in computer vision. In video surveillance, for example, our methodology allows for the detection of objects in a cluttered background. We show how the methodology can be adapted to simultaneously align a batch of images and correct serious defects/corruptions in each image, opening new perspectives.

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